Calculus Examples

$\frac{d y}{d x} = \frac{1}{2} \sqrt{y} \cos^{2} (\sqrt{y})$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})}$ .

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} (\frac{1}{2} \sqrt{y} \cos^{2} (\sqrt{y}))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1}{2} \cdot \frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \sqrt{y} \cos^{2} (\sqrt{y})$

Step 1.2.2

Combine.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{2 (\sqrt{y} \cos^{2} (\sqrt{y}))} \sqrt{y} \cos^{2} (\sqrt{y})$

Step 1.2.3

Cancel the common factor of $\sqrt{y}$ .

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Step 1.2.3.1

Factor $\sqrt{y}$ out of $2 (\sqrt{y} \cos^{2} (\sqrt{y}))$ .

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{\sqrt{y} (2 (\cos^{2} (\sqrt{y})))} \sqrt{y} \cos^{2} (\sqrt{y})$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{\sqrt{y} (2 (\cos^{2} (\sqrt{y})))} \sqrt{y} \cos^{2} (\sqrt{y})$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{2 (\cos^{2} (\sqrt{y}))} \cos^{2} (\sqrt{y})$

Step 1.2.4

Cancel the common factor of $\cos^{2} (\sqrt{y})$ .

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Step 1.2.4.1

Factor $\cos^{2} (\sqrt{y})$ out of $2 (\cos^{2} (\sqrt{y}))$ .

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{\cos^{2} (\sqrt{y}) \cdot 2} \cos^{2} (\sqrt{y})$

Step 1.2.4.2

Cancel the common factor.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{\cos^{2} (\sqrt{y}) \cdot 2} \cos^{2} (\sqrt{y})$

Step 1.2.4.3

Rewrite the expression.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1 \cdot 1}{2}$

Step 1.2.5

Multiply $1$ by $1$ .

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} \frac{d y}{d x} = \frac{1}{2}$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} d y = \frac{1}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y} \cos^{2} (\sqrt{y})} d y = \int \frac{1}{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Multiply by $1$ .

$\int \frac{1}{\sqrt{y} (\cos^{2} (\sqrt{y}) \cdot 1)} d y = \int \frac{1}{2} d x$

Step 2.2.1.2

Separate fractions.

$\int \frac{1}{\sqrt{y} \cdot (1)} \cdot \frac{1}{\cos^{2} (\sqrt{y})} d y = \int \frac{1}{2} d x$

Step 2.2.1.3

Convert from $\frac{1}{\cos^{2} (\sqrt{y})}$ to $\sec^{2} (\sqrt{y})$ .

$\int \frac{1}{\sqrt{y} \cdot (1)} \sec^{2} (\sqrt{y}) d y = \int \frac{1}{2} d x$

Step 2.2.1.4

Multiply $\sqrt{y}$ by $1$ .

$\int \frac{1}{\sqrt{y}} \sec^{2} (\sqrt{y}) d y = \int \frac{1}{2} d x$

Step 2.2.1.5

Combine $\frac{1}{\sqrt{y}}$ and $\sec^{2} (\sqrt{y})$ .

$\int \frac{\sec^{2} (\sqrt{y})}{\sqrt{y}} d y = \int \frac{1}{2} d x$

Step 2.2.2

Apply basic rules of exponents.

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Step 2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\int \frac{\sec^{2} (y^{\frac{1}{2}})}{\sqrt{y}} d y = \int \frac{1}{2} d x$

Step 2.2.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\int \frac{\sec^{2} (y^{\frac{1}{2}})}{y^{\frac{1}{2}}} d y = \int \frac{1}{2} d x$

Step 2.2.2.3

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int \sec^{2} (y^{\frac{1}{2}}) {(y^{\frac{1}{2}})}^{- 1} d y = \int \frac{1}{2} d x$

Step 2.2.2.4

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sec^{2} (y^{\frac{1}{2}}) y^{\frac{1}{2} \cdot - 1} d y = \int \frac{1}{2} d x$

Step 2.2.2.4.2

Combine $\frac{1}{2}$ and $- 1$ .

$\int \sec^{2} (y^{\frac{1}{2}}) y^{\frac{- 1}{2}} d y = \int \frac{1}{2} d x$

Step 2.2.2.4.3

Move the negative in front of the fraction.

$\int \sec^{2} (y^{\frac{1}{2}}) y^{- \frac{1}{2}} d y = \int \frac{1}{2} d x$

Step 2.2.3

Let $u = y^{\frac{1}{2}}$ . Then $d u = \frac{1}{2 y^{\frac{1}{2}}} d y$ , so $- d u = \frac{- 1}{2 y^{\frac{1}{2}}} d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.3.1

Let $u = y^{\frac{1}{2}}$ . Find $\frac{d u}{d y}$ .

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Step 2.2.3.1.1

Differentiate $y^{\frac{1}{2}}$ .

$\frac{d}{d y} [y^{\frac{1}{2}}]$

Step 2.2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} y^{\frac{1}{2} - 1}$

Step 2.2.3.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.2.3.1.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.2.3.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.2.3.1.6

Simplify the numerator.

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Step 2.2.3.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} y^{\frac{1 - 2}{2}}$

Step 2.2.3.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} y^{\frac{- 1}{2}}$

Step 2.2.3.1.7

Move the negative in front of the fraction.

$\frac{1}{2} y^{- \frac{1}{2}}$

Step 2.2.3.1.8

Simplify.

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Step 2.2.3.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{y^{\frac{1}{2}}}$

Step 2.2.3.1.8.2

Multiply $\frac{1}{2}$ by $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{1}{2 y^{\frac{1}{2}}}$

Step 2.2.3.2

Rewrite the problem using $u$ and $d u$ .

$\int 2 \sec^{2} (u) d u = \int \frac{1}{2} d x$

Step 2.2.4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int \sec^{2} (u) d u = \int \frac{1}{2} d x$

Step 2.2.5

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$2 (\tan (u) + C_{1}) = \int \frac{1}{2} d x$

Step 2.2.6

Simplify.

$2 \tan (u) + C_{1} = \int \frac{1}{2} d x$

Step 2.2.7

Replace all occurrences of $u$ with $y^{\frac{1}{2}}$ .

$2 \tan (y^{\frac{1}{2}}) + C_{1} = \int \frac{1}{2} d x$

Step 2.3

Apply the constant rule.

$2 \tan (y^{\frac{1}{2}}) + C_{1} = \frac{1}{2} x + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$2 \tan (y^{\frac{1}{2}}) = \frac{1}{2} x + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 \tan (y^{\frac{1}{2}}) = \frac{1}{2} x + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 \tan (y^{\frac{1}{2}}) = \frac{1}{2} x + K$ by $2$ .

$\frac{2 \tan (y^{\frac{1}{2}})}{2} = \frac{\frac{1}{2} x}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 \tan (y^{\frac{1}{2}})}{2} = \frac{\frac{1}{2} x}{2} + \frac{K}{2}$

Step 3.1.2.1.2

Divide $\tan (y^{\frac{1}{2}})$ by $1$ .

$\tan (y^{\frac{1}{2}}) = \frac{\frac{1}{2} x}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Combine $\frac{1}{2}$ and $x$ .

$\tan (y^{\frac{1}{2}}) = \frac{\frac{x}{2}}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\tan (y^{\frac{1}{2}}) = \frac{x}{2} \cdot \frac{1}{2} + \frac{K}{2}$

Step 3.1.3.1.3

Multiply $\frac{x}{2} \cdot \frac{1}{2}$ .

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Step 3.1.3.1.3.1

Multiply $\frac{x}{2}$ by $\frac{1}{2}$ .

$\tan (y^{\frac{1}{2}}) = \frac{x}{2 \cdot 2} + \frac{K}{2}$

Step 3.1.3.1.3.2

Multiply $2$ by $2$ .

$\tan (y^{\frac{1}{2}}) = \frac{x}{4} + \frac{K}{2}$

Step 3.2

Substitute $u$ for $y^{\frac{1}{2}}$ .

$\tan (u) = \frac{x}{4} + \frac{K}{2}$

Step 3.3

Reorder $\frac{x}{4}$ and $\frac{K}{2}$ .

$\tan (u) = \frac{K}{2} + \frac{x}{4}$

Step 3.4

Take the inverse tangent of both sides of the equation to extract $u$ from inside the tangent.

$u = \arctan (\frac{K}{2} + \frac{x}{4})$

Step 3.5

Substitute $y^{\frac{1}{2}}$ for $u$ and solve $y^{\frac{1}{2}} = \arctan (\frac{K}{2} + \frac{x}{4})$

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Step 3.5.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {\arctan (\frac{K}{2} + \frac{x}{4})}^{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{2} \cdot 2} = {\arctan (\frac{K}{2} + \frac{x}{4})}^{2}$

Step 3.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {\arctan (\frac{K}{2} + \frac{x}{4})}^{2}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

$y^{1} = {\arctan (\frac{K}{2} + \frac{x}{4})}^{2}$

Step 3.5.2.1.2

Simplify.

$y = {\arctan (\frac{K}{2} + \frac{x}{4})}^{2}$

Step 4

Simplify the constant of integration.

$y = {\arctan (K + \frac{x}{4})}^{2}$